Robustness and limitations of fractal-dimension estimation methods for complex fluid interfaces

Fractal geometry provides a useful framework for characterizing highly wrinkled interfaces arising in fluid-dynamical and combustion processes. In this work, the robustness of three methods for estimating the fractal dimension, Df D f , of empirical fractals is assessed: a filtering-based approach, the correlation dimension, and a grid-based box-counting method. The methods are applied to two challenging three-dimensional DNS datasets, namely primary liquid-jet atomization with strong droplet formation and a flame interface subject to Richtmyer-Meshkov instability (RMI). While all three methods show excellent agreement for mathematically defined fractals and reasonable agreement for weakly wrinkled surfaces, they are pushed to their limits for the complex physical interfaces considered here. Differences arise from both numerical sensitivities and the distinct structural and textural features captured by the individual estimators, and may lead to values outside expected or even physical bounds. Although the inferred inner cut-off scales remain within plausible ranges, method-dependent differences persist. For jet atomization, the box-counting method deviates markedly from the other approaches, whereas for the RMI configuration the filtering-based method is the clear outlier. Overall, the results demonstrate that heterogeneous empirical interfaces do not possess a unique, method-independent fractal dimension. Among the investigated methods, the correlation-dimension approach proved to be the most robust and consistently yielded physically plausible estimates across the wide range of interface topologies considered.